#### Vol. 29, No. 1, 1969

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Triple series equations involving Laguerre polynomials

### John S. Lowndes

Vol. 29 (1969), No. 1, 167–173
##### Abstract

In this paper it is shown that the problem of solving the triple series equations of the first kind

 ∑ n=0∞A nΓ(α + 1 + n)Ln(α;x) = 0, 0 ≦ x < a, (1) ∑ n=0∞A nΓ(α + β + n)Ln(α;x) = f(x), a < x < b, (2) ∑ n=0∞A nΓ(α + 1 + n)Ln(α;x) = 0, b < x < ∞, (3)
and the triple series equations of the second kind
 ∑ n=0∞B nΓ(α + β + n)Ln(α;x) = g(x), 0 ≦ x < a, (4) ∑ n=0∞B nΓ(α + 1 + n)Ln(α;x) = 0, a < x < b, (5) ∑ n=0∞B nΓ(α + β + n)Ln(α;x) = h(x), b < x < ∞, (6)
where α + β > 0, 0 < β < 1, Ln(α;x) = Lnα(x) is the Laguerre polynomial and f(x), g(x) and h(x) are known functions, can be reduced to that of solving a Fredholm integral equation of the second kind. The analysis is formal and no attempt is made to supply details of rigour.

Primary: 45.20
Secondary: 33.00
##### Milestones 